4. Including the Mass of the Spring. In practice, all springs have mass. We will use a model for a spring with mass to calculate the kinetic energy of the spring itself. We know that the mass has a kinetic energy of Mvz. What about the spring? We will make two basic assumptions to form a model that will allow us to find the kinetic energy of the spring. Figure 2: A Spring with Mass First assumption: y = 0'_ ' This end has The mass per unit length ofthe spring is uniform. '7 = 0 If we assume the spring has a mass, MS and a length L, then any section of the spring of length f would have its ma i n b . Piece of SS g V6 y' spring of I" dy thickness dy g = & (or) at location y L _ M5 This end m _ f T (9) / also moves y = L" at 12m lvm Second assumption: The speed of the coils of the spring varies linearly with position. The bottom of the spring and the hanging mass have the same speed, 12m, while the top of the spring has a speed of zero. Furthermore, the speed of a particular part of the spring varies linearly with the location on the spring: v(y) =%vm ([0) Determine the mass, din, of a piece of the spring that has a thickness of dy. This expression should be in terms of Ms, L, and dy. dm= Determine the kinetic energy, dK, of a piece of the spring at location 31 with a thickness dy, as shown in Figure 2. This equation should have Ms, L, dy, y, and 11m in it. dK = Now use the integral to nd the total kinetic energy of the spring itself: K = foLdK Kspring =